\chapter{Basic Theory}

This equations and derivations in this chapter and the next chapter is mainly based on a working note by Ivar Ertesvåg \cite{Ertesvåg2004}.

\section{Transport equations}
The momentum equation is given as
%Momentum equation
\begin{equation}
\rho \frac{D{{u}_{i}}}{Dt}=\frac{\partial }{\partial t}(\rho {{u}_{i}})+\frac{\partial }{\partial {{x}_{j}}}(\rho {{u}_{i}}{{u}_{j}})=\frac{\partial }{\partial {{x}_{j}}}(-p{{\delta }_{ij}}+{{\tau }_{ij}})+\rho {{f}_{i}},
\label{eq:momentum}
\end{equation}

Where $\tau_{ij}$ is the viscous stress tensor and $f_i$ represents body force. 

In a mixture of different compounds the continuity equation can be expressed for each species $k$ and we get the mass fraction 

\begin{equation}
\frac{\partial}{\partial t} (\rho Y_k) + \frac{\partial}{\partial x_j} ( \rho Y_k u_j) = \frac{\partial}{\partial x_j} ( -j_{k,j})+R_k
\label{eq:massfrac}
\end{equation}

where $j_{k,j}$ is the mass flux and $R_k$ is the reaction rate.

The continuity equation is obtained from summing Eq. \ref{eq:massfrac} for all species
%continuity equation
\begin{equation}
\frac{D \rho}{Dt}+\rho \frac{\partial u_j}{\partial x_j}=0
\label{eq:Continuity eq}
\end{equation}

and the energy equation is given as

%Energy equation
\begin{equation}
\rho \frac{D e}{D t}=\frac{\partial}{\partial x_j} \left( -q_j - \sum_k{h_k j_{k,j}} \right)- p \frac{\partial u_j}{\partial x_j} + \tau_{ij} \frac{\partial u_i}{\partial x_j} + Q + F
\label{eq:Energy equation}
\end{equation}

where the term $Q$ is the volumetric energy production, and $F$ is the energy source term due to body forces.


\section{Flux relations}
This section gives the relations for different molecular fluxes.

The Newtonian viscous stress tensor is given as
%Newtonian viscous stress tensor
\begin{equation}
\tau_{ij}=\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)+(\mu_B-\frac{2}{3}\mu)\frac{\partial u_l}{\partial x_l}\delta_ij.
\label{eq:visc stress tensor}
\end{equation}

Fouriers law for heat transfer

%heat transfer
\begin{equation}
q_j=-\lambda \frac{\partial T}{\partial x_j}
\label{eq:Fouriers law}
\end{equation}


With the assumption that mass diffusion is controlled by the concentration gradients only, and not by the temperature or pressure gradients, we get Fick's law of mass diffusion
%Fick's Law
\begin{equation}
-j_{k,j}=\rho \mathcal{D}_k \frac{\partial Y_k}{\partial x_j}
\label{eq:Ficks law}
\end{equation}

Where $\mathcal{D}_k$ is the diffusivity of compound $k$ in the mixture.


\begin{equation}
-q_j-\sum_k{h_k j_{k,j}}=\frac{\lambda}{c_p} \frac{\partial h}{\partial x_j}-\sum_k{ \left(\frac{\lambda}{c_p}-\rho \mathcal{D}_k \right) h_k \frac{\partial Y_k}{\partial x_j}}.
\label{eq:enthalpy gradient}
\end{equation}

Here the thermal conductivity and the specific heat capacity is for the mixture, while the diffusion coefficient is for the individual species.

\section{Entropy equation}
%Hentet fra Entropy production and Transport in Turbulent flames from DNS
The chemical potential for ideal gases is equal to the specific Gibbs function $\mu_k=g_k=h_k-Ts_k$. From Ertesvåg and Kolbu(2005) \cite{Ertesvåg2005} the classical Gibbs equation can be expressed 

\begin{equation}
T \rho \frac{Ds}{Dt}=\rho \frac{De}{Dt}-\frac{p}{\rho} \frac{D \rho}{D t}-\rho \sum_k{\mu_k \frac{D Y_k}{Dt}}.
\label{eq:classical Gibbs}
\end{equation}

The entropy equation is then obtained from the above equation and the equations of mass, continuity, and energy

\begin{multline}
\rho \frac{Ds}{Dt}= \frac{\partial}{\partial x_j} \left(-\frac{q_j}{T}-\sum_k{s_{k} j_{k,j}} \right)-\left(-\frac{q_j}{T^2}\right)\frac{\partial T}{\partial x_j}+ \frac{1}{T} \sum_k{(-j_{k,j}) \left( \frac{\partial \mu_k}{\partial x_j} \right)_T }\\ 
+\frac{Q}{T}+\frac{\Phi}{T}+\frac{F}{T}-\frac{1}{T} \sum_k{\mu_k R_k}.
\label{eq:Entropy equation}
\end{multline}

$Q$ is the volumetric energy production, either by internal sources or by radiation. $\Phi$ is the viscous dissipation term. F is a energy source term due to body forces

The equation for mean entropy generation per unit volume (here taken from \cite{Ertesvåg2005}) was first derived by Hirschfelder et.al \cite{Hirschfelder1954}. It kan be derived from the turbulence averaging relations shown in the next chapter.

\begin{multline}
\frac{\partial}{\partial t}(\overline{\rho}\tilde{s}) + \frac{\partial}{\partial x_j}(\overline{\rho} \tilde{s} \tilde{u}_j) = \frac{\partial}{\partial x_j} \left( \overline{-\frac{q_j}{T}-\sum_k {s_k j_{k,j}}}-\overline{\rho s'' u_j''} \right) \\
+ \overline{ \left(-\frac{q_j}{T^2} \right) \frac{\partial T}{\partial x_j}} + \overline{\sum_k{\left(- \frac{j_{k,j}}{T} \right)} \left(\frac{\partial \mu_k}{\partial x_j} \right)_{T}} \\
+ \overline{\left(\frac{Q}{T}\right)} + \overline{\left(\frac{\Phi}{T}\right)}+\overline{\left(\frac{F}{T}\right)} - \overline{\left(\frac{1}{T} \sum_k {\mu_k R_k}\right)}
\label{eq:meanentropy}
\end{multline}


The radiation term $\frac{Q}{T}$ and the body force $\frac{F}{T}$ is set to zero in order to simplify the calculations. The terms $\overline{ \left(-\frac{q_j}{T^2} \right) \frac{\partial T}{\partial x_j}}, \overline{\sum_k{\left(- \frac{j_{k,j}}{T} \right)} \left(\frac{\partial \mu_k}{\partial x_j} \right)_{T}}, - \overline{\left(\frac{1}{T} \sum_k {\mu_k R_k}\right)}$ and $ \overline{\left(\frac{\Phi}{T}\right)}$ are respectively the entropy production terms due to heat transfer, mass transfer, chemical reaction and viscous dissipation of the flow. 

\section{Turbulence and density averaged equations}

The Reynolds number is given as the ratio between inertia forces and viscous forces in the flow. It is a dimensionless value whch is important for flow characterization. For pipe flow the Reynolds number is given by Eq. \ref{eq:Reynolds number} where $\nu$ is the dynamic viscosity.

\begin{equation}
Re_d=\frac{Ud}{\nu}
\label{eq:Reynolds number}
\end{equation}

When the Reynolds number surpasses a critical value, the flow becomes unsteady and a transition from laminar to turbulent flow may happen. The critical Reynolds number for pipe flow is typically around 2300. If Reynolds number higher than 10000, the flow is considered to be fully turbulent. If $Re_d$ is somwhere in between the flow is in transition. 

A turbulent flow is characterized by chaotic movement and random fluctuations. The characteristics of the flow are very different from those of a laminar flow. The flow contains rotating eddies of different length and time scales. By separating the flow into a mean value and a fluctuating value the equations of momentum, mass and energy can be more easily be rewritten to a manageable form which accounts for the properties of the turbulent flow.

Reynolds averaging gives

%Reynolds averaging
\begin{equation}
\phi=\overline{\phi}+\phi'
\label{eq:Reynoldsturbdecomp}
\end{equation}

where $\phi$ is a general property. The mean component $\overline{\phi}$ is defined as the average value of the property over a long time interval $\Delta t$ as in

\begin{equation}
\overline{\phi}=\int_{t_0}^{t_0+\Delta t}{\phi(t) dt}
\label{eq:Reynoldsave}
\end{equation}

and $\phi'$ is its fluctuation.

In a turbulent flame there are large fluctuations in density, species concentrations, flow velocity and temperature. Favre-averaging (density-averaging) is therefore the preferred method of describing reacting flows.

%Favre averaging
\begin{equation}
\phi =\tilde{\phi }+{\phi }''
\label{eq:Favre}
\end{equation}

here $\tilde{\phi }$ is averaged by the density

\begin{equation}
\tilde{\phi }=\frac{\overline{\rho \phi }}{{\bar{\rho }}}
\label{eq:Favreturbdecomp}
\end{equation}

and ${\phi }''$ is it's fluctuation.

\subsection{Turbulence averaged equations}

The turbulent kinetic energy is given as

\begin{equation}
k=\frac{1}{2}\overline{u_i' u_i'}=\frac{1}{2}(\overline{u_1'^2}+\overline{u_2'^2}+\overline{u_3'^2})
\label{eq:turbulent kinetic energy}
\end{equation}

The Favre averaged continuity equation is
\begin{equation}
\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial}{\partial x_j}(\overline{\rho} \tilde{u}_j)=0.
\label{eq:Favrecont}
\end{equation}

Inserting Eq. \ref{eq:Favre} in to Eq. \ref{eq:momentum} yields the Favre averaged momentum equation, Eq. \ref{eq:Favre average momentum}.

%Favre averaged momentum equation
\begin{equation}
\frac{\partial}{\partial t}(\overline{\rho} \tilde{u_i}) + \frac{\partial}{\partial x_j}(\overline{\rho} \tilde{u_i} \tilde{u_j})=-\frac{\partial \overline{p}}{\partial x_i}+\frac{\partial}{\partial x_j}(\overline{\tau_{ij}}-\overline{\rho u_i'' u_j''}+\overline{\rho f_i}).
\label{eq:Favre average momentum}
\end{equation}

The Favre averaged energy equation is

\begin{equation}
\frac{\partial }{\partial t}(\bar{\rho }\tilde{h})+\frac{\partial }{\partial {{x}_{j}}}(\bar{\rho }\tilde{h}{{{\tilde{u}}}_{j}})=\frac{\partial \bar{p}}{\partial t}+{{{\tilde{u}}}_{j}}\frac{\partial \bar{p}}{\partial {{x}_{j}}}+\overline{{u}''\frac{\partial p}{\partial {{x}_{j}}}}+\frac{\partial }{\partial {{x}_{j}}}\left( -{{{\bar{q}}}_{j}}-\sum\limits_{k}{\overline{{{h}_{k}}{{j}_{k,j}}}}-\overline{\rho {{h}^{\prime \prime }}{{{{u}''}}_{j}}} \right) \\ 
+\bar{\Phi }+\bar{Q}
\label{eq:Favreenergy}
\end{equation}

The transport equation for the mean mass fraction is
\begin{equation}
\frac{\partial }{\partial t}(\bar{\rho }{{\tilde{Y}}_{k}})+\frac{\partial }{\partial {{x}_{j}}}(\bar{\rho }{{\tilde{Y}}_{k}}{{\tilde{u}}_{j}})=\frac{\partial }{\partial {{x}_{j}}}(-\overline{{{j}_{k,j}}}-\overline{\rho {{{{Y}''}}_{k}}{{{{u}''}}_{j}}})+{{\bar{R}}_{k}}
\label{eq:Favremass}
\end{equation}

For an ideal gas

\begin{equation}
\frac{1}{T} \left(\frac{\partial \mu_k}{\partial x_j} \right)_T=\mathcal{R}_k \left(\frac{1}{Y_k}\frac{\partial Y_k}{\partial x_j}+\frac{1}{p}\frac{\partial p}{\partial x_j} \right)
\label{eq:simplified potential}
\end{equation}

The pressure term in \ref{eq:simplified potential} can be set to 0 in a jet diffusion flame since variation in pressure are minimal compared to the concentration gradients.